<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><font color="Maroon" title="c3">I</font> =  the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font>;<br/>
<a NAME="E1:77_1_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a>  the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="77_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:77_1_1_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a>  the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:77_1_1_1"/><span class="kw">then </span>
 the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 
;<br/>
<a NAME="E3:77_1_1_1"/><span class="kw">then </span>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a>  the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#T67" onmouseover="rs('zmodul02/T67')" onmouseout="rh()">ZMODUL02:67</a></span>;<br/>
<a NAME="E4:77_1_1_1"/><span class="kw">then </span>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#D3" onmouseover="rs('vectsp_7/D3')" onmouseout="rh()">VECTSP_7:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:77_1_1_1"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:77_1_1"/><span class="kw">then </span>
 not  the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:77_1_1"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">v</font> = <font color="Maroon" title="c4">v</font> as   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77_1_1"><span class="lab"><font color="Green" title="E40">A1</font></span></a></span>;<br/>
<a NAME="E6:77_1_1"/>
( <font color="Maroon" title="c1">V</font> is   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c1">V</font> &amp;  the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  &amp;  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a>  the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T32" onmouseover="rs('zmodul01/T32')" onmouseout="rh()">ZMODUL01:32</a>, <a class="ref" href="vectsp_7.html#D3" onmouseover="rs('vectsp_7/D3')" onmouseout="rh()">VECTSP_7:def 3</a></span>;<br/>
<a NAME="E7:77_1_1"/><span class="kw">then </span>
(  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"> the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"> the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c1">V</font> &amp;  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"> the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> is  <a href="vectsp_7.html#V3" title="VECTSP_7:attr.3">free</a>  &amp;  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="vectsp_7.html#V3" title="VECTSP_7:attr.3">free</a>  &amp; <font color="Maroon" title="c5">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77_1_1"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="zmodul06.html#T24" onmouseover="rs('zmodul06/T24')" onmouseout="rh()">ZMODUL06:24</a></span>;<br/>
<a NAME="E8:77_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A6</font></span>: 
 not <font color="Maroon" title="c5">v</font> is  <a href="zmodul06.html#V1" title="ZMODUL06:attr.1">torsion</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul06.html#D3" onmouseover="rs('zmodul06/D3')" onmouseout="rh()">ZMODUL06:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">pp</font> = <font color="Maroon" title="c2">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a> ;<br/>
<span class="kw">assume </span><a NAME="E10:77_1_1"/>
 <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c1">V</font></span>)</span>) is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="p1">(<span class="default"><a href="zmodul07.html#K2" title="ZMODUL07:func.2">ZQMorph</a> (<font color="Maroon" title="c1">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c1">V</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c1">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c1">V</font></span>)</span>)</span>)</span>

<br/>.= 
 <a href="vectsp10.html#K4" title="VECTSP10:func.4">zeroCoset</a> (<font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c1">V</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#D6" onmouseover="rs('vectsp10/D6')" onmouseout="rh()">VECTSP10:def 6</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c1">V</font></span>)</span>

;<br/>
<a NAME="E12:77_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c1">V</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul07.html#D2" target="_self" onmouseover="rs('zmodul07/D2')" onmouseout="rh()">defMophVW</a></span>;<br/>
<a NAME="E13:77_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T63" onmouseover="rs('zmodul01/T63')" onmouseout="rh()">ZMODUL01:63</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">u</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E15:77_1_1"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">pp</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c7">u</font>
 ;<br/>
<span class="p1">(<span class="default"> the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> = 
 the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" onmouseover="rs('xboole_1/T39')" onmouseout="rh()">XBOOLE_1:39</a></span>
<br/>.= 
 the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77_1_1"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a>, <a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>
;<br/>
<a NAME="E17:77_1_1"/><span class="kw">then </span>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a>  the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"> the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#T72" onmouseover="rs('zmodul02/T72')" onmouseout="rh()">ZMODUL02:72</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">u1</font>, <font color="Maroon" title="c9">u2</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E19:77_1_1"/><span class="lab"><font color="Green" title="E43">B4</font></span>: 
( <font color="Maroon" title="c8">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"> the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &amp; <font color="Maroon" title="c9">u2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">u2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T92" onmouseover="rs('zmodul01/T92')" onmouseout="rh()">ZMODUL01:92</a>, <a class="ref" href="zmodul03.html#T14" onmouseover="rs('zmodul03/T14')" onmouseout="rh()">ZMODUL03:14</a></span>;<br/>
<a NAME="E20:77_1_1"/><span class="lab"><font color="Green" title="E44">B5</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">pp</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c8">u1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">pp</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">u2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:77_1_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="txt" href="#E19:77_1_1"><span class="lab"><font color="Green" title="E43">B4</font></span></a>, <a class="ref" href="vectsp_1.html#D14" onmouseover="rs('vectsp_1/D14')" onmouseout="rh()">VECTSP_1:def 14</a></span>;<br/>
<a NAME="E21:77_1_1"/><span class="lab"><font color="Green" title="E45">B6</font></span>: 
<font color="Maroon" title="c6">pp</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c8">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"> the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:77_1_1"><span class="lab"><font color="Green" title="E43">B4</font></span></a>, <a class="ref" href="zmodul01.html#T37" onmouseover="rs('zmodul01/T37')" onmouseout="rh()">ZMODUL01:37</a></span>;<br/>
<a NAME="E22:77_1_1"/><span class="lab"><font color="Green" title="E46">B7</font></span>: 
<font color="Maroon" title="c6">pp</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">u2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:77_1_1"><span class="lab"><font color="Green" title="E43">B4</font></span></a>, <a class="ref" href="zmodul01.html#T37" onmouseover="rs('zmodul01/T37')" onmouseout="rh()">ZMODUL01:37</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">iu2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a><span class="kw"> such that </span><br/><a NAME="E24:77_1_1"/><span class="lab"><font color="Green" title="E47">B8</font></span>: 
<font color="Maroon" title="c9">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">iu2</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:77_1_1"><span class="lab"><font color="Green" title="E43">B4</font></span></a>, <a class="ref" href="zmodul06.html#T19" onmouseover="rs('zmodul06/T19')" onmouseout="rh()">ZMODUL06:19</a></span>;<br/>
<a NAME="E25:77_1_1"/><span class="lab"><font color="Green" title="E48">B9</font></span>: 
<font color="Maroon" title="c6">pp</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">pp</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">iu2</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:77_1_1"><span class="lab"><font color="Green" title="E47">B8</font></span></a>, <a class="ref" href="vectsp_1.html#D16" onmouseover="rs('vectsp_1/D16')" onmouseout="rh()">VECTSP_1:def 16</a></span>;<br/>
<a NAME="E26:77_1_1"/><span class="lab"><font color="Green" title="E49">B10</font></span>: 
<font color="Maroon" title="c1">V</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"> the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77_1_1"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="zmodul04.html#T33" onmouseover="rs('zmodul04/T33')" onmouseout="rh()">ZMODUL04:33</a></span>;<br/>
<a NAME="E27:77_1_1"/><span class="lab"><font color="Green" title="E50">B11</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>
 
;<br/>
<a NAME="E28:77_1_1"/><span class="lab"><font color="Green" title="E51">B12</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul06.html#T20" onmouseover="rs('zmodul06/T20')" onmouseout="rh()">ZMODUL06:20</a></span>;<br/>
<a NAME="E29:77_1_1"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"> the   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c1">V</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T33" onmouseover="rs('zmodul01/T33')" onmouseout="rh()">ZMODUL01:33</a></span>;<br/>
<a NAME="E30:77_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">pp</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">iu2</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">pp</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">iu2</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">pp</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">iu2</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:77_1_1"><span class="lab"><font color="Green" title="E44">B5</font></span></a>, <a class="txt" href="#E27:77_1_1"><span class="lab"><font color="Green" title="E50">B11</font></span></a>, <a class="txt" href="#E21:77_1_1"><span class="lab"><font color="Green" title="E45">B6</font></span></a>, <a class="txt" href="#E22:77_1_1"><span class="lab"><font color="Green" title="E46">B7</font></span></a>, <a class="txt" href="#E25:77_1_1"><span class="lab"><font color="Green" title="E48">B9</font></span></a>, <a class="txt" href="#E26:77_1_1"><span class="lab"><font color="Green" title="E49">B10</font></span></a>, <a class="txt" href="#E28:77_1_1"><span class="lab"><font color="Green" title="E51">B12</font></span></a>, <a class="ref" href="zmodul01.html#T134" onmouseover="rs('zmodul01/T134')" onmouseout="rh()">ZMODUL01:134</a></span>;<br/>
<a NAME="E31:77_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">pp</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">iu2</font></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">pp</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">iu2</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">pp</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">iu2</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T9" onmouseover="rs('zmodul01/T9')" onmouseout="rh()">ZMODUL01:9</a></span>;<br/>
<a NAME="E32:77_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">pp</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">iu2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:77_1_1"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="ref" href="rlvect_1.html#T15" onmouseover="rs('rlvect_1/T15')" onmouseout="rh()">RLVECT_1:15</a></span>;<br/>
<a NAME="E33:77_1_1"/><span class="kw">then </span>
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">p</font> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>;<br/>
<span class="kw">hence </span><a NAME="E34:77_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</a>, <a class="ref" href="nat_d.html#T33" onmouseover="rs('nat_d/T33')" onmouseout="rh()">NAT_D:33</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
